Nuffield Free-Standing Mathematics Activity Stationary points Maximum point Minimum point

Stationary points Think about Maximum point point of inflexion Think about What happens to the gradient as the curve passes through a maximum point? a minimum point? a point of inflexion? Minimum point

Stationary points At a maximum point + – At a minimum point + – = 0 = 0 + – is negative At a minimum point = 0 – + is positive

Stationary points At some stationary points + + and These are: = 0 + = 0 and These are: points of inflexion – – Note is also 0 at some maximum and minimum points

To sketch y = 5 + 4x – x2 When x = 0, y = 5 = 4 – 2x When y = 0, At stationary points = 0 5 + 4x – x2 = 0 2x = 4 ( - x)( + x) = 0 5 1 x = 2 x = 5 or x = – 1 = – 2 x y y = 5 + 4x – x2 2 9 The stationary point is a maximum 5 –1 y = 5 + 4 2 – 22 = 9 There is a maximum point at (2, 9)

To sketch y = 2 + 3x2 – x3 When x = 0, y = 2 = 6x – 3x2 When y = 0, 2 + 3x2 – x3 = 0 At stationary points this is 0 y = 2 + 3x2 – x3 6x – 3x2 = 0 y x maximum (2,6) 3x = 0 (2 – x) x = 0 or x = 2 minimum (0, 2) = 6 – 6x When x = 0 this is positive When x = 2, is negative and y = 2 and y = 2 + 12 – 8 = 6 Minimum point (0, 2) Maximum point (2, 6)

To sketch y = x4 – 4 = 4x3 When x = 0, y = – 4 At stationary points = 0 When y = 0, 4x3 = 0 x4 – 4 = 0 x4 = 4 x = 0 y = – 4 x2 = 2 x = ± 2 = 12x2 = 0 x y y = x4 – 4 Gradient before x = 0 is negative Gradient after x = 0 is positive Ö2 – 4 – Ö2 There is a minimum point at (0, – 4)

Stationary points Reflect on your work How does finding stationary points help you to sketch a curve? Is it possible to know how many stationary points there are by just looking at the function? What other information is it useful to find before you attempt to sketch a curve?